The effect of the nutation angle on the flow inside a precessing cylinder is experimentally explored and compared with numerical simulations. The focus is laid on the typical breakdown of the directly forced m = 1 Kelvin mode for increasing precession ratio (Poincaré number) and the accompanying transition between laminar and turbulent flows. Compared to the reference case with a 90° nutation angle, prograde rotation leads to an earlier breakdown, while in the retrograde case, the forced mode continues to exist also for higher Poincaré numbers. Depending largely on the occurrence and intensity of an axisymmetric double-roll mode, a kinematic dynamo study reveals a sensitive dependence of the self-excitation condition on the nutation angle and the Poincaré number. Optimal dynamo conditions are found for 90° angle which, however, might shift to slightly retrograde precession for higher Reynolds numbers.
Precession-induced fluid motion is essential in various phenomena and applications, including fuel payloads of rotating spacecraft,1,2 atmospheric vortices like tornadoes and hurricanes, and the flow in the Earth's liquid outer core.3 The suggestion that precessional forcing can act as a potential power source for Earth's magnetic field4 had initiated a debate on the non-trivial issue of the energy budget required for the geodynamo. The early argument that a precessing laminar flow cannot convert enough energy to maintain Earth's magnetic field5,6 changes completely for turbulent flows, which can dissipate more energy, thereby making it possible to sustain the geomagnetic field.7 Meanwhile, precession is also believed to be responsible for the dynamos of the ancient moon8,9 and the asteroid Vesta,10 and several numerical studies have evidenced that magnetic fields indeed can be generated via precession-driven flows.11–14
In the meantime, a variety of experiments focusing on precession-driven flows were conducted in numerous laboratories.15–22 The closest to a hydromagnetic dynamo was that of Gans23 who performed a precession-driven liquid metal experiment in 1971 and achieved a magnetic field amplification by a factor of three. This experiment motivated the development of a large-scale precession dynamo experiment, which is presently under construction at Helmholtz–Zentrum Dresden–Rossendorf (HZDR) within the framework of the DRESDYN project.24 After the success of the pioneering liquid sodium experiments in Riga,25 Karlsruhe,26 and Cadarache,27 the DRESDYN experiment aims at achieving homogeneous dynamo action without the use of propellers, pumps, guiding tubes, or magnetic materials. Prior studies had indeed demonstrated that precession can act as an efficient mechanism for driving an intense flow in a homogeneous fluid.28
The DRESDYN precession experiment consists of a cylinder with a radius of R = 1 m and a height of H = 2 m. The cylinder rotates around its symmetry axis at the cylinder frequency fc ≤ 10 Hz. It is mounted on a rotating turntable, which precesses the cylinder at the precession frequency fp ≤ 1 Hz. With this rotation rate of the cylinder, we can achieve up to [with the Reynolds number defined in Eq. (2)]. In this experiment, liquid sodium will be used as a working fluid to accomplish dynamo action.24,29 To understand the dynamics of the flow for the large-scale experiment, a 1:6 down-scaled water test experiment with the same aspect ratio and rotation rates has been in operation at HZDR for many years.22,30,31
Precession produces complex three-dimensional flow structures as a consequence of the interaction among inertial modes, boundary layers, and the directly driven base flow.22,29,32 In the cylindrical geometry, the inertial modes are also called Kelvin modes.33 Each mode has its own eigen-frequency, which is determined by the radial (n), axial (k), and azimuthal (m) wave numbers.34 The Reynolds number Re, the precession ratio or Poincare number (), the geometric aspect ratio ( H/R), and the nutation angle α (angle between the axis of precession and the axis of rotation ) determine the flow inside the precessing cylinder35 (see also Fig. 1).
This study examines the influence of different nutation angles on the dominant flow modes inside the precessing cylinder for prograde and retrograde rotation configurations. For this purpose, we conduct water experiments and compare their results with numerical simulations.32 Focusing on the inertial modes with the largest energy fractions, we use direct measurements of the axial velocity with ultrasound Doppler velocimetry (UDV) and simulation data from a three-dimensional numerical model. Finally, we utilize the obtained flow fields in a kinematic dynamo model in order to identify the most promising parameter range for dynamo action in the upcoming DRESDYN experiment. In this kinematic dynamo model, the flow is prescribed using the full information obtained from numerical simulations of the hydrodynamic problem, whereas at this stage any feedback of the field on the flow via the Lorentz force is still ignored.
This paper is structured as follows: in Sec. II, we briefly describe the setup and the employed measurement procedure for the down-scaled water experiment. The theory and numerics are presented in Sec. III, including both the hydrodynamic and the kinematic dynamo problem. Our findings from hydrodynamic experiments and simulations are described and compared in Sec. IV. The results of dynamo action for different configurations are then discussed in Sec. V. In Sec. VI, the results are summarized, and some prospects for the future work and the large-scale liquid sodium experiment are discussed.
II. DOWN-SCALED WATER PRECESSION EXPERIMENT
A. Experimental setup
Figure 1(a) gives a picture of the 1:6 down-scaled water precession experiment. The experiment comprises a water-filled acrylic cylindrical vessel with radius R = 163 mm and height H = 326 mm. The vessel is connected with an asynchronous 3 kW motor via a transmission chain, which allows us to adjust the rotation rate of the cylinder. The rotational frequency of the cylinder fc can reach a maximum of 10 Hz. The cylinder's end caps are joined axially by eight rods to keep their alignment in parallel. This entire system is mounted on a turntable powered by a second 2.2 kW asynchronous motor, which can rotate up to a frequency fp of 1 Hz. Both rotation rates, i.e., fc and fp, are continuously measured by two tachometers and recorded by a data acquisition system. The nutation angle α formed by the cylinder rotation axis and the turntable rotation axis can be varied between 60° and 90°. The present study conducts experiments for the three different nutation angles of α = 60°, 75°, and 90°.
B. Measurement procedure
To determine the flow field inside the cylinder, ultrasonic transducers (TR0408SS, Signal Processing SA, Lausanne) are placed at one end cap of the cylinder [Fig. 1(a)]. They are connected with an ultrasound Doppler velocimeter (UDV) (Dop 3010, Signal Processing SA, Lausanne), which measures the velocity with a temporal resolution of 0.94 Hz. Each transducer emits an ultrasonic pulse and receives echoes reflected from particles in the path of the ultrasonic beam on a regular basis. The velocimeter then infers the axial flow velocity in front of the sensor from the Doppler shift of the recorded echoes.
The UDV sensor is mounted 150 mm from the center of one end cap as shown in Fig. 1(a) and measures the full height of the cylinder (i.e., H = 326 mm). For all measurements, the interrogated volume per sample measures 0.97 mm in the longitudinal direction, 2.5 mm lateral at 10 mm from the sensor, and 15.9 mm lateral at the opposite end cap.
The tracer particles used in the water inside the cylindrical vessel consist of a mixture of Griltex 2A P1 particles with sizes of 50 μm (60% by weight) and 80 μm (40% by weight). These tracer particles have a density of 1.02 g/cm3. Before commencing the experimental run, the water is vigorously mixed at a high rotation rate to ensure that the tracer particles are distributed evenly throughout the cylindrical vessel. The vessel is then set into rotation at fc until the fluid co-rotates with the vessel, which is indicated by a vanishing velocity on the UDV channel. We then set the turntable in motion at fp and waited for more than 50 periods of cylinder rotation until a steady state was attained. The velocity profile is then captured for 52 cylinder rotations. Finally, we increase the precession rate to the next value of fp and wait until the fluid motion at the increased precession rate has reached a steady state. This measurement protocol is consistent with other experimental studies of precession-driven flows for different geometries.36–39
Initially, the measurement started at a value of Po = 0.035 (i.e., Hz), which was then gradually increased in steps up to Po = 0.198 (i.e., Hz). The flow measurements were conducted with the ultrasonic transducer at radius r = 150 mm and at a constant rotation rate of the cylinder Hz, corresponding to for Po = 0.1 and a kinematic viscosity of . These experiments were carried out with two different rotation directions, i.e., prograde and retrograde precession. During the experiments, room temperature was kept as constant as possible using air-conditioning, thereby eliminating the temperature-dependent viscosity as a possible influence on the flow.
III. THEORY AND NUMERICS
We study an incompressible fluid of kinematic viscosity ν enclosed in a cylindrical container of radius R and height H, which rotates with angular velocity of () for prograde (retrograde) motion and precesses with angular velocity of . The nutation angle α is measured from the precession to the rotation axes, as illustrated in Fig. 1(b). Another option is considering α to run between 0° and 180°; however, we use the range between 0° and 90° and distinguish between pro- and retrograde precession.
The governing equation, which describes the fluid motion inside the precessing cylinder, is the Navier–Stokes equation32,40,41
together with the incompressibility condition . Here, u is the velocity flow field, is the total rotation vector, and r is the position vector with respect to the center of the cylinder. (force per unit mass) is the gradient of the reduced pressure, which incorporates the hydro-static pressure and the gradient terms such as the centrifugal force. P does not change the dynamical behavior of the flow. The two terms on the right-hand side, which contain the rotation vector , are the Coriolis and the Poincaré forces, respectively. This Poincaré force is also called Euler force.42 z, r, and denote the axial, radial, and azimuthal cylindrical coordinates, respectively. Equation (1) is complemented by no-slip boundary conditions at the walls. In order to non-dimensionalize the Navier–Stokes equation, we use the radius R as the length scale and as the timescale. The latter choice relies on using the projection of the total angular velocity on the cylinder axis, i.e., .
The dimensionless parameters governing the problem are the Reynolds number Re, the Poincaré number Po, and the container aspect ratio Γ, which read, respectively,
The governing equation for the spatiotemporal evolution of the magnetic field B is the (dimensionless) induction equation
For the time-averaged velocity field , the solution of the linear evolution equation has the form with σ representing the eigenvalue. In Eq. (3), Rm is the magnetic Reynolds number, defined as
where η is the magnetic diffusivity of the fluid.
C. Numerical methods
Precession-driven flows in cylinders can be simulated in one of the following two frames of reference: (i) the mantle frame (attached to the cylinder wall) or (ii) the turntable frame in which the cylinder walls rotate at Ωc and the total vector is stationary. We select the latter for which so that the Poincaré force vanishes. While for the hydrodynamic problem, we use a spectral element Fourier code43 (meshing the domain with 300 quadrilateral elements in the meridional half plane and 128 Fourier modes in the azimuthal direction) with no-slip boundary conditions, we solve the induction equation through a finite volume method (see Appendix A) with constraint transport in order to satisfy . For the dynamo simulations, we apply pseudo-vacuum boundary conditions for the magnetic field (only tangential components vanish at the wall). Comparable studies have been applied to other dynamo experiments,44–46 from which it is known that pseudo-vacuum boundary conditions lead to a lower threshold for the occurrence of a dynamo compared to more realistic insulating boundary conditions. For the models of the VKS dynamo, deviations in the order of 30% were found,47 while for the Riga dynamo, a deviation of approximately 15% was found.48
The simulation protocol is as follows: we start at t = 0 with a pure solid body rotation motion, which in the turntable reference frame is ,40 imposing forcing by precession. Once the statistically steady regime is achieved by the hydrodynamic flow field, we average in time and put this flow structure in the induction equation (3). The kinematic dynamo simulations are run until more than one diffusion time, when the exponential growth or decay in the simulation is established.
The phase space investigated for the kinematic dynamo problem ranges between and the Poincaré number in the range ±. The nutation angles are , both for prograde and retrograde precession, as well as for 90°. The aspect ratio will be fixed at Γ = 2.
This is rather close to the resonant case, which can be computed from the requirement that the forcing frequency matches the eigenfrequency of the inertial wave, which is given by
where λmkn plays the role of a radial wave number. In our system with the timescale given by the inverse of the rotation frequency, the resonance condition translates to , and Eq. (5) can be solved for Γ in combination with the dispersion relation for inertial waves, which reads
The collection of all simulations in the parameter space (Re, Po) will be shown later in Fig. 4.
A. Typical flow patterns
In this subsection, we present the results of the precession water experiment and the flow pattern obtained from simulations. The contour plots of the flow structures obtained are shown in Fig. 2. These plots illustrate the evolution of the axial velocity profile uz over time t and depth z. (the depth indicates the distance along the transducer's axis from the transducer). The first row displays the axial velocity uz in comparison between simulation and experiment at α = 60° (prograde), while the second row shows the velocity contour from simulation at α = 60° (prograde). The third through seventh rows display experimental results for α = 60°, 75°, and 90° for both prograde and retrograde precession. The dominating oscillatory pattern of the velocity profile, defined by the rotational frequency Ωc of the cylinder, represents the standing inertial mode with as recorded by the rotating UDV sensor mounted on the vessel wall.
As we increase fp, the response of the fluid begins to change. At lower values of Po (see Fig. 2, the first column), the flow pattern exhibits a stable flow structure and is almost vertical in both cases (prograde and retrograde). However, as Po exceeds a certain higher value for α = 60° (prograde), α = 75° (prograde), and 90°, as shown in the third column of Figs. 2(b3), 2(c3), 2(d3), and 2(e3), the flow pattern changes and exhibits a breaking of the equatorial symmetry. In contrast, in the retrograde case [see Figs. 2(f3) and 2(g3)], it remains almost unchanged, and the occurrence of the equatorial symmetry breaking is shifted to larger values of Po. The breaking of the equatorial symmetry indicates a flow state transition at a critical value of the precession ratio, implying the presence of Kelvin modes with even axial wave numbers.29 In addition, the transition that occurred at the critical Po has a considerable effect on the amplitudes of the Kelvin modes, as we will demonstrate in the following.
B. Quantitative results and comparison with numerics
In a more quantitative analysis, the mode amplitudes are calculated by decomposing the axial velocity field into (m, k) modes. For both the simulations and the experiments, we start by applying a discrete sine transformation (DST) to the axial velocity for each time step. This is done in the following manner for our numerical simulations:
where zl is the discrete axial coordinate defined by . Then this intermediate result is further processed by means of a 2D-FFT in time and azimuth29
Here, r represents the radial position of a virtual sensor, which is set to 150 mm from the center (similar to the location of the UDV probe in the experiment), and and tn are the azimuthal coordinates and times from the simulations. Equation (8) represents the ω-dependent amplitude for modes with fixed m and k for numerical simulations.
In contrast, for the experimentally recorded data, the radius is fixed so that . The axial DST for the k-modes becomes
Now, the data are processed in chunks of individual cylinder rotations, where j0 and Nj mark the index of the first sample and the number of samples in each rotation, respectively. Then, we employ the DFT (discrete Fourier transformation) again to obtain the (m, k) modes
With this decomposition, various modes can be observed inside the precessing cylinder. In our analysis, we examined only those modes that have substantial amplitudes and are most relevant for dynamo action,31 i.e., the mode, directly excited by precessional forcing, and the mode, which is also called axisymmetric oscillation,49,50 because in an unforced inviscid and rotating fluid, this mode exhibits a periodic behavior with the frequency .
However, in the present case, the axisymmetric oscillation with is forced as a result of a non-linear interaction of the directly forced mode with itself.51,52 This non-linear self-interaction of the forced mode [which beside the wave numbers is additionally characterized by a frequency ω = 1 due to the forcing] can drive two velocity components, which are of the form with ω = 0 and with ω = 2, respectively. Here, we discuss the first solution, which has the geometrical structure of an axially symmetric oscillatory eigenmode, but due to the special form of generation, it has the frequency zero. The second possibility is supposed to be suppressed because of the boundary conditions at the end caps.
Figure 3 shows the amplitudes of the prominent modes vs the precession ratio for both cases (prograde and retrograde) at 60°, 75°, and 90°, respectively. In order to compare with the experimental data, the simulated flow amplitudes at Re = 6500 were linearly extrapolated to the experimental value . The reason for this procedure lies in the lower limit of the experimental facility in terms of cylinder rotation frequency, which corresponds to and the extremely time-consuming behavior of simulations at . The linear extrapolation is motivated by the rather close value of the Reynolds numbers and our previous experimental results, which show a quasi-linear scaling with Re for the flow velocity.29 For prograde 60°, 75°, and 90° [see Figs. 3(a), 3(b), and 3(c)], we observe that the directly forced mode increases up to , beyond which there is an abrupt transition of the flow state due to the breakdown of the mode (see Appendix B). Simultaneously, an axially symmetric mode appears in a narrow range of Po, which corresponds to the double roll structure that was previously shown to be most relevant for dynamo action.29 Note that for prograde 60° and 75° at higher Po, we observe a discrepancy between the results of simulation and experiment for the mode. This could be due to the concentration of double roll near the wall boundary where this flow structure is computed. Remarkably, the nutation angle influences the critical Po, such that as the angle increases (60°, 75°, and 90°), so does the critical Po (0.083, 0.087, and 0.10). In other words, for smaller nutation angles, the transition sets in earlier.
In contrast, the data for the retrograde 60° and 75° cases show no clear breakdown of the directly forced mode , which has a gradual decrease in amplitude. At the same time, we observe a smoother increase in the axially symmetric mode within the considered range of Po, as shown in Figs. 3(d) and 3(e). In comparison to all other cases, (retrograde) exhibits the largest amplitude of the mode, and (retrograde) has the smallest. As compared to the prograde case, the critical Po values are shifted to larger values for the retrograde case. The slight offsets of the experimental data with respect to the numerical data along the x-axis are probably due to the difference in Re: as Re increases, its critical Po decreases slightly.32 In general, the experimental values at for two different configurations (prograde and retrograde) and at 90° are in good agreement with the results of the numerical simulations with the exception of the case 75° prograde when Po is large.
V. DYNAMO RESULTS
In this section, we present the results of the kinematic dynamo code applied to the flow fields as obtained in Sec. IV. The criterion for finding dynamo action is an exponential increase in the volume-averaged magnetic energy. From the temporal evolution of the magnetic energy, the growth rate is calculated.
The analysis will focus on two main points: (i) the influence of the nutation angle α on the ability to drive dynamo action; (ii) the impact of the Reynolds number for a fixed angle .
A. The role of the nutation angle
Figure 4 shows the regime diagram in the space at fixed Re = 6500. We find dynamo action (red symbols) for all the angles except for prograde. The range of the precession ratio where dynamo action occurs changes with the nutation angle: for prograde cases, dynamos occur at while for retrograde they appear at Po >0.15 with a more extended range. For each angle, the blue diamonds indicate the dynamo with the largest growth rate.
We plot the growth rate of the magnetic field in Fig. 5. As already highlighted, the case prograde shows no positive growth rate even at the largest magnetic Reynolds number considered here. The lowest critical magnetic Reynolds number occurs for which, therefore, turns out to be the most promising case for the later dynamo experiment. Also the magnetic field structure, in this case the azimuthal component , depends on α (Fig. 6). The three snapshots are taken between t = 300 and 380. Both cases exhibit contours elongated along the axis, and the final field shows a change in sign during the evolution.
B. The role of the Reynolds number
In this subsection, we fix the nutation angle to investigate the impact of the hydrodynamic Reynolds number on the flow regime and the dynamo action. We select this angle since presently it appears to be the best angle for dynamo action with the lowest critical magnetic Reynolds number.
Figure 7 shows the regime diagram in the (Re, Po) space where the meaning of the symbols is consistent with that of Fig. 4: black squares denote no dynamo action, red triangles indicate dynamo action, and the blue diamond signifies the strongest dynamo action. The blue curve is a fit marking the scaling for the maximum growth rate. Notice that for Re < 3500, we observe dynamos also significantly above the threshold curve; in contrast for larger Re, the dynamo action is restricted in a quite narrow range. Due to the limited range of Re, the scaling is just a crude indication for the best precession ratio in terms of dynamo action.
Since the complete formula of the blue curve is , we expect an asymptotic behavior for a large Reynolds number. This expectation is supported by the fact that the best values of Po for dynamo are characterized by the strongest poloidal flow, i.e., , which has also been observed to scale asymptotically with a similar exponent.29 This flow structure is reminiscent to the s2t1 pattern,53 which is known from the study of Dudley and James54 to be able to excite a dynamo at rather low values of the magnetic Reynolds number. This s2t1 pattern consists of two poloidal rolls (here, the double roll structure as for example visualized in Fig. 11 in Appendix C) and one toroidal structure (here the azimuthal circulation that is essentially independent of z).
Notice the presence of an isolated area of dynamo action for Re = 8000 and Po = 0.075 (for Rm > 900). We currently do not have an explanation for this.
In the next step, we select the best precession ratio for every Re (the blue diamonds) and show the growth rate γ as a function of Rm in Fig. 8(a). The slopes of the curves seem to converge for the highest Reynolds number considered here. Collecting the points where the lines cross the γ = 0 we plot the critical magnetic Reynolds number in Fig. 8(b). The trend is not monotonic, showing a flat maximum in the range . The smallest critical magnetic Reynolds number is found for Re = 2000. This might be the case since at small Reynolds, the flow tends to remain well organized in large scale structures rather than becoming turbulent with the presence of small scales.
VI. CONCLUSION AND PROSPECTS
In this study, we investigated the effects of different nutation angles (60°, 75°, and 90°) on a precession-driven flow in the cylindrical geometry for both prograde and retrograde motion. We compared experimental results to direct numerical simulations. Experimentally, the axial flow uz was measured by a UDV sensor mounted on the end cap of the cylinder. These velocities were decomposed into several (m, k) modes. We chose and modes for our study, because they have significant amplitudes and are relevant for dynamo studies. In all cases, the experimental results agreed well with the numerical findings. For prograde precession with α = 60°, 75°, and for 90°, the flow abruptly transitions from a laminar regime to a turbulent regime, which goes along with the sudden decrease in the directly forced flow. In contrast, retrograde motion does not show a clear breakdown of the directly forced mode , but rather a smooth increase in the axisymmetric mode . In this regard, the different behavior of the between prograde and retrograde motion could be explained in the context of the non-linear theory of Waleffe55 and specifically for the precessing cylinder by Meunier:52 the Kelvin mode characterized by will produce, through non-linear interaction with itself, a flow of the form (0, 2). Since the (1, 1) profiles for retrograde cases do not show an abrupt breakdown and the peaks are broader, it follows that also the (0, 2) mode undergoes a similar behavior. The close linkage of these modes can also be observed in Fig. 10 ( Appendix B) where the decrease in (1, 1) coincides with an increase in the (0, 2) mode indicating non-linear interaction.
However, there is currently no quantitative explanation for the peculiar behavior of the amplitude of the mode. Applicable possibilities might be the influence of boundary layer phenomena,32 the destabilization of the circulation flow due to the non-axisymmetric forced flow,29 or the geostrophic instability.56 The simulations also suggest that the emergence of the mode cannot be exclusively attributed to a centrifugal instability,29,32 because in all cases, the maximum amplitude of is reached for centrifugally stable configurations of the flow.32 The tendency of retrograde precession to provide stronger large-scale flow amplitude without a breakdown of the directly forced mode should be interesting for dynamo purposes because, in principle, this allows an injection of more energy into the flow without breaking the base flow. A reason for different properties of the time-averaged large-scale flow might be a preference of the excitation of inertial waves57 when these waves focus onto wave attractors, which may occur when the inertial waves are reflected at boundaries that are neither parallel nor perpendicular to the rotation axis.58 With this question in mind, we further investigated whether the (time-averaged) flow fields obtained from the hydrodynamic simulations are capable of driving a dynamo. We conducted kinematic dynamo simulations, which can be summarized according to the following points:
The nutation angle α is crucial both for the hydrodynamic flow structure and the resulting dynamo action. In the phase diagram (Fig. 4), we show that (at the present state) the most efficient dynamo (with the highest growth rate γ) is found at . The reason for that lies in the rich and optimal flow structure for this nutation angle.32,41 With a view on Fig. 5, it is tempting to assume that a slightly retrograde motion might provide an even lower threshold for the onset of dynamo action.
Interestingly, the case does not show any dynamo action. The reason may be in the lower value of the flow amplitude with respect to the other nutation angles, as shown in Fig. 3(a). In other words, the higher value of goes along with a low value of the and vice versa, preventing a synergy effect between them.
For the particular case , the hydrodynamic Reynolds number slightly affects the best precession ratio range where dynamo action is found. This precession ratio scales as . At low Re, the dynamos occur in a broader range of Po more extended than for larger Reynolds, e.g., for Re = 2000. The critical magnetic Reynolds number shows a weak dependence on Re with a slight increase around but approaches the previously known value of 430 when going to . Given that the real dynamo experiment can achieve an Rm value of 700, there seems to be a reasonable safety margin to reach dynamo action. However the extrapolation to the hydrodynamic regime of the DRESDYN precession experiment must be considered with a grain of salt and has to be carefully checked in the larger experiment.
The present work can be extended in several directions. The possibility of the down-scaled water experiment to reach Reynolds numbers of up to 2 × 106 should be utilized to confirm the −1/4 scaling of the critical precession ratio also for nutation angles different from 90°. From the numerical point of view, there is the possibility to use stress-free boundary condition for the velocity on the end-caps in order to check the specific impact of those end wall's boundary layers. The kinematic dynamo code should be extended to the use of vacuum boundary conditions which might still lead to some increase in the critical Rm when compared to the presently used vertical field conditions. Finally, in a more advanced study, the fully coupled system of induction and Navier–Stokes equations, including the back-reaction of the Lorentz forces, should be investigated. For our precession system, with its very sensitive dependence on various parameters, this fully non-linear system promises to show particularly interesting effects.
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Program (Grant Agreement No. 787544) and from Deutsche Forschungsgemeinschaft under Project No. GI 1405/1-1.
Conflict of Interest
The authors have no conflicts to disclose.
Vivaswat Kumar: Conceptualization (supporting); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (equal); Software (equal); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Federico Pizzi: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (equal); Software (lead); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). André Giesecke: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Resources (lead); Software (equal); Supervision (lead); Validation (lead); Writing – review & editing (equal). Ján Simkanin: Software (supporting); Writing – review & editing (supporting). Thomas Gundrum: Resources (equal); Supervision (equal); Validation (supporting). Matthias Ratajczak: Conceptualization (supporting); Formal analysis (supporting); Investigation (supporting); Supervision (equal); Visualization (supporting); Writing – review & editing (equal). Frank Stefani: Conceptualization (lead); Data curation (supporting); Funding acquisition (lead); Methodology (lead); Project administration (lead); Supervision (lead); Visualization (supporting); Writing – original draft (equal); Writing – review & editing (equal).
The data that support the findings of this study are available from the corresponding author upon reasonable request.
APPENDIX A: COMPARISON BETWEEN THE SPECTRAL-FOURIER METHOD AND THE FINITE VOLUME METHOD
The hydrodynamic DNS code provides data on a non-uniform meridional grid (where the nodal coordinates are given by the zeros of a polynomial of degree eight used to approximate the solution within a spectral element). Since a Fourier decomposition is used in the azimuthal direction, the distribution of the -coordinates is uniform, therefore, for the finite volume method we only interpolated in the meridional plane (r, z). The interpolation is done linearly by making use of triangles formed by Delaunay triangulation. For the finite volume scheme, we used a spatial resolution of 400 grid cells along the axis, 200 grid cells in the radial direction, and 32 in the azimuthal direction. The solution of the interpolated flow field (as used as input for the solver of the induction equation) is almost the same as that from the hydrodynamic code, as shown in Fig. 9 for the three velocity components. As we can observe both the meridional and the polar contours coincide well, proving that a reduced number of the azimuthal Fourier mode is enough.
APPENDIX B: FLOW TRANSITION
Figure 10 depicts the flow behavior before and after the transition for (prograde). At lower Po until 0.081, the flow appears stable and is dominated by the directly forced mode. During the transition, i.e., at Po = 0.087, begins to rise. This phenomenon is evident when examining the contour plots of Figs. 10(b) and 10(c), where Fig. 10(b) depicts the plot immediately following a change in Po from . Figures 10(a) and 10(b) show nearly identical patterns, whereas Fig. 10(c) depicts a significant change in the flow structure. This is the definitive sign of a flow state transition and is consistent with previous published results (experimentally and numerically29,31).
APPENDIX C: THE TIME AVERAGED FLOW FIELD
The time-averaging of the velocity field obtained in the simulations is performed in the precession frame of reference. In that frame, the directly driven flow, which is essentially the inertial mode with , is a standing wave so that these contributions do not cancel when integrating in time. Figure 11 visualizes the total flow (left) and the axisymmetric contributions (right) for one paradigmatic case ( and ). Note that the axisymmetric flow shown on the right hand side is included in the total flow on the left and provides a perturbation that (among other contributions) gives rise to the breaking of the particular equatorial symmetry.